Derived Demand for Healthcare

Derived Demand for Healthcare as an Input to Producing Health

Few people want a hospital stay, a blood test or a course of physiotherapy for the experience itself. Health economics treats such services as inputs that people combine with their own time to produce better health, so the amount of care they seek follows from how much health they want and how costly health is to produce. Michael Grossman formalised the argument, deriving a demand function for medical care from the demand for health with implications for age, education, wages and prices, including the possibility, shown in an illustrative calculation below, that care use rises while desired health falls. Empirical tests support parts of the idea and expose its limits. The full model belongs to the Grossman model page, and the focus here is the step from health to care.

From wanting health to buying care

Kenneth Arrow observed in 1963 that medical services, apart from preventive services, give satisfaction only in the event of illness, which he could find true of hardly any other significant item in the average budget. Grossman's later review of his own work restates a point he credits to students of medical economics, that what consumers demand when they buy medical services is better health, not the services as such, a distinction he says had not been exploited in formal work before 1972.

The tool that turns this observation into a model is household production. Consumers are treated as producing the commodities that enter their utility function from market goods and their own time, in the way a firm produces output from inputs. Grossman draws the conclusion directly. Since goods and services are inputs into producing commodities, the demand for them is a derived demand for a factor of production, so the demand for medical care and other health inputs is derived from the basic demand for health.

In Grossman's framework the produced commodity is a durable health capital stock. Grossman notes that diet, housing, smoking and excessive alcohol use also affect health, the last two with negative marginal products, and that the model keeps its structure whatever the main purchased health input is.

How a demand function for care follows from the demand for health

Because health is produced, its price is not the fee for a consultation. Grossman calls the cost of health capital its shadow price, which depends on many variables besides the price of medical care. Health demanded falls as the shadow price rises, and shifts in the variables behind it alter the derived demand for health inputs.

Three conditions give the demand for the health stock a constant-elasticity form. The first is the pure investment version of the model, in which healthy time gives no utility directly, the second is a zero real interest rate, as in Grossman's empirical formulation, and the third is his healthy-time function $h_t = 8760 - B H_t^{-C}$, in which healthy time $h_t$ rises with the stock towards the 8,760 hours in a year ($B$ and $C$ are positive constants). Under these conditions the demand for health is as follows.

$$\ln H_t = \varepsilon \ln W_t - \varepsilon \ln \pi_t - \varepsilon \ln \delta_t$$

where $H_t$ is the stock of health at age $t$, $W_t$ is the wage rate, $\pi_t$ is the marginal cost of producing gross investment in health, $\delta_t$ is the rate of health depreciation, $\varepsilon$ is the elasticity of the marginal efficiency of capital schedule (how strongly the health stock demanded responds to its cost), and intercepts are suppressed. Gross investment equals depreciation plus net investment, so when net investment is small, gross investment is approximately $\delta_t H_t$.

Medical care enters through the health production function for gross investment, which Grossman specifies in Cobb-Douglas form with medical care and own time as inputs. Combining the demand for health, the investment identity and cost-minimising input choice gives his reduced-form demand function for medical care, shown below.

$$\ln M_t = [(1-K)\varepsilon + K] \ln W_t - [(1-K)\varepsilon + K] \ln P_t + \rho_H(\varepsilon - 1)E + (1-\varepsilon)\ln \delta_t$$

where $M_t$ is medical care, $P_t$ is the price of medical care, $K$ is the share of own time in the total cost of gross investment, $E$ is the stock of knowledge measured by years of schooling, $\rho_H$ is the proportional rise in the marginal products of care and own time from one more unit of $E$, and the other terms are as defined above. As in Grossman's estimating equation, a small term in the rate of net disinvestment is omitted.

Every coefficient in the demand for care contains $\varepsilon$, the parameter of the demand for health, which is the formal content of "derived". The price coefficient has two parts, in which $(1-K)\varepsilon$ is a scale effect, because dearer care raises the shadow price of health and less health is demanded, and $K$ is a substitution effect towards own time.

What the derived demand predicts for age, education, wages and prices

The derived-demand logic produces predictions that a simple price and quantity model of care would miss, some against intuition. The table compares effects on health and on care demanded in the pure investment model.

Grossman argues that $\varepsilon$ is likely to be below one. Because healthy time is bounded by 8,760 hours a year, his healthy-time function gives $\varepsilon = 1/(1+C)$, which is below one since $C$ is positive.

The wage prediction changes where care is free at the point of use and rationed by waiting and travel time. The price of care then becomes the wage multiplied by the hours needed to obtain it, and Grossman shows that with a third, time-free input in health production the wage elasticity of medical care can be zero or negative, while a higher wage still raises the health stock demanded.

Worked example in which care use rises as desired health falls

The figures are illustrative. A person behaves as in the pure investment model with $\varepsilon = 0.5$, a zero real interest rate, an unchanged wage and price of care, and gross investment approximately equal to $\delta H$.

1. Depreciation doubles. The depreciation rate rises from 2% to 4% a year between two ages. The demand function for health gives the following.

$$\frac{H_1}{H_0} = \left(\frac{0.04}{0.02}\right)^{-0.5} = \frac{1}{\sqrt{2}} \approx 0.707$$

where $H_0$ and $H_1$ are the health stocks demanded before and after the rise. The person now wants about 29% less health capital.

2. Gross investment rises. Gross investment is depreciation times the stock, as below.

$$\frac{I_1}{I_0} = \frac{0.04}{0.02} \times 0.707 \approx 1.414$$

where $I_0$ and $I_1$ are gross investment before and after the rise. Twice the depreciation rate on a stock 29% smaller needs about 41% more investment.

3. Medical care rises with investment. With the wage and the price of care unchanged, the cost-minimising mix of care and own time is unchanged, so care rises by the same 41%, as the demand function for care confirms, since $(1 - 0.5) \times \ln 2 \approx 0.347$, whose exponential is about 1.414.

4. The direction depends on the elasticity. If $\varepsilon$ were 1.5, the stock would fall to $H_1/H_0 = 2^{-1.5} \approx 0.354$ and gross investment, twice the depreciation rate on that smaller stock, to $I_1/I_0 = 2^{-0.5} \approx 0.707$, so care would fall by about 29%.

5. One more year of schooling. Suppose an extra year of schooling raises the marginal products of care and own time by 6% ($\rho_H = 0.06$). With $\varepsilon = 0.5$, health demanded changes by $\rho_H \varepsilon = 0.06 \times 0.5 = 0.03$, a rise of 3%, and medical care by $\rho_H(\varepsilon - 1) = 0.06 \times (0.5 - 1) = -0.03$, a fall of 3%.

6. Price response. If own time accounts for 40% of the cost of gross investment ($K = 0.4$), the price elasticity of demand for care is $-(0.6 \times 0.5 + 0.4) = -0.7$, made up of $-0.3$ through the lower health stock demanded and $-0.4$ from substitution towards own time.

7. Interpretation. The older person uses more care and has less health; the better-educated person has more health and uses less care. Sicker people using more care is consistent with derived demand, and care use alone is a poor guide to how much health a person wants.

Testing the derived demand against data

Tests estimate demand functions for health and care and check the signs against the model. In Grossman's own estimates from a nationally representative 1963 United States survey, age significantly raised medical expenditure while reducing health, as predicted, but the wage and schooling coefficients in the care equation had signs contrary to the model's predictions, though neither was significant.

The sharpest test concerns the relation between health and care. Holding health fixed in a conditional demand function for care, the model predicts a positive health coefficient, since more health demanded means more inputs demanded. Grossman reports that Wagstaff, using the 1976 Danish Welfare Survey, instead found it negative and very significant, and according to Grossman, Zweifel and Breyer took this as grounds for rejecting the idea that spending on care is a demand derived from the demand for health. Grossman answers that the estimate is biased when health is entered as if it were fixed. People whose health depreciates faster both buy more care and have less health, which pushes the coefficient below its true value.

Health production-function estimates, which face the same bias, point the same way. Grossman reports that Stratmann, using the 1989 United States National Health Interview Survey with work-loss days due to illness in the past two weeks as the health measure, found that people with a doctor visit had more work-loss days in single-equation models but fewer once the probability of a visit was treated as endogenous, with estimated reductions of 2.7 days for influenza, 2.9 days for impairments and 6.9 days for chronic asthma.

Where the derived-demand account falls short

The framework rests on assumptions that health care often breaks. Grossman ruled out uncertainty in his formal development, whereas Arrow described demand for medical services as irregular and unpredictable; later theory, Grossman concludes, finds larger expected health stocks and health inputs under uncertainty.

Arrow also noted that the cost of care depends on the doctor's willingness to use medical services and that widespread insurance increases the demand for care, so observed demand reflects the principal-agent problem in healthcare and moral hazard as well as the demand for health.

The framework says nothing on its own about how productive care is, and Grossman cites evidence that medical care may be an unimportant determinant of health in developed countries.

Derived demand in economic evaluation and health policy

The derived-demand view supports judging care by the health it produces. Cost-utility analysis counts outcomes in health units such as the QALY, and the derived-demand argument explains why counts of consultations or bed days measure inputs rather than outputs.

It also affects how demand data are read. A utilisation rate can rise because a population is ageing and its health depreciating faster, with no change in preferences or in the underlying demand function for health, and indeed with less health demanded. When charges change, the response of care includes a change in the health demanded, so the price elasticity of healthcare is partly a property of the demand for health. Demand forecasting can rely on observed demand, but judging whether extra use is worthwhile needs evidence on its health effects.

Derived demand beside its neighbouring concepts

Several pages cover parts of the same chain from health to care. The table sets out what each one holds.

Common misreadings of derived demand

Several errors recur when the idea is applied. One is to read a negative association between good health and care use as evidence against derived demand, when people with faster depreciation are predicted to hold less health and buy more care. Another is to treat a rise in care use as a rise in the demand for health. A third is to assume that derived demand means patients choose care well, which uncertainty, agency and insurance all undermine.

Sources

• Arrow KJ. Uncertainty and the welfare economics of medical care. American Economic Review. 1963;53(5):941-973. https://www.aeaweb.org/aer/top20/53.5.941-973.pdf
• Grossman M. The Demand for Health: A Theoretical and Empirical Investigation. NBER Occasional Paper 119. New York: National Bureau of Economic Research; 1972. Chapter 1. https://www.nber.org/books-and-chapters/demand-health-theoretical-and-empirical-investigation
• Grossman M. The human capital model. In: Culyer AJ, Newhouse JP, eds. Handbook of Health Economics. Vol 1A. Amsterdam: Elsevier; 2000:347-408. https://doi.org/10.1016/S1574-0064(00)80166-3